MinT - Best Known (52, 52+28, s)-Nets in Base 16

Best Known (52, 52+28, s)-Nets in Base 16

(52, 52+28, 583)-Net over F₁₆ — Constructive and digital

Digital (52, 80, 583)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 20, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]
2. digital (32, 60, 518)-net over F₁₆, using
  - trace code for nets [i] based on digital (2, 30, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(52, 52+28, 594)-Net in Base 16 — Constructive

(52, 80, 594)-net in base 16, using

1 times m-reduction [i] based on (52, 81, 594)-net in base 16, using
- (u,Â u+v)-construction [i] based on
  1. (9, 23, 80)-net in base 16, using
    - 1 times m-reduction [i] based on (9, 24, 80)-net in base 16, using
      - base change [i] based on digital (1, 16, 80)-net over F₆₄, using
        net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
  2. digital (29, 58, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 29, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(52, 52+28, 3192)-Net over F₁₆ — Digital

Digital (52, 80, 3192)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁸⁰, 3192, F₁₆, 28) (dual of [3192, 3112, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁸⁰, 4103, F₁₆, 28) (dual of [4103, 4023, 29]-code), using
  - constructionÂ X applied to C_e(27) âŠ‚ C_e(25) [i] based on
    1. linear OA(16⁷⁹, 4096, F₁₆, 28) (dual of [4096, 4017, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
    2. linear OA(16⁷³, 4096, F₁₆, 26) (dual of [4096, 4023, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
    3. linear OA(16¹, 7, F₁₆, 1) (dual of [7, 6, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(16¹, s, F₁₆, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(52, 52+28, 3062323)-Net in Base 16 — Upper bound on s

There is no (52, 80, 3062324)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 135993 690588 603205 710145 274595 767831 368951 617658 109425 519686 329427 044601 049558 546270 503573 124916 > 16⁸⁰ [i]